View source for Bernoulli's principle

{{Short description|Relates pressure and flow velocity in fluid dynamics}}
{{about| Bernoulli's principle and Bernoulli's equation in fluid dynamics|Bernoulli's theorem in probability|law of large numbers|an unrelated topic in [[ordinary differential equation]]s|Bernoulli differential equation}}
[[File:VenturiFlow.png|right|thumb|A flow of air through a [[venturi meter]]. The kinetic energy increases at the expense of the [[fluid pressure]], as shown by the difference in height of the two columns of water.]]
[[File:Venturi Tube en.webm|thumb|Video of a [[Venturi effect|venturi meter]] used in a lab experiment]]
{{Continuum mechanics|fluid}}

In [[fluid dynamics]], '''Bernoulli's principle''' states that an increase in the speed of a fluid occurs simultaneously with a decrease in static [[pressure]] or a decrease in the [[fluid]]'s [[potential energy]].<ref name="Clancy1975">{{cite book |last=Clancy |first=L.J. |authorlink=Laurence Joseph Clancy |title=Aerodynamics |url=https://books.google.com/books?id=zaNTAAAAMAAJ |year=1975 |publisher=Wiley |isbn=978-0-470-15837-1}}</ref>{{rp|at=Ch.3}}<ref name="Batchelor2000">{{cite book |last=Batchelor |first=G.K. |authorlink=George Batchelor |title=An Introduction to Fluid Dynamics |url=https://books.google.com/books?id=Rla7OihRvUgC&pg=PA156 |year=2000 |publisher=[[Cambridge University Press]] |location=Cambridge |isbn=978-0-521-66396-0}}</ref>{{rp|at= § 3.5|pp=156–164}} The principle is named after [[Daniel Bernoulli]] who published it in his book ''[[Hydrodynamica]]'' in 1738.<ref>{{cite web | url =http://www.britannica.com/EBchecked/topic/658890/Hydrodynamica#tab=active~checked%2Citems~checked&title=Hydrodynamica%20–%20Britannica%20Online%20Encyclopedia | title=Hydrodynamica | accessdate=2008-10-30 |publisher= Britannica Online Encyclopedia }}</ref> Although Bernoulli deduced that pressure decreases when the flow speed increases, it was [[Leonhard Euler]] who derived '''Bernoulli's equation''' in its usual form in 1752.<ref name="Anderson2016">{{citation | title=Handbook of fluid dynamics | edition=2nd | editor-last=Johnson | editor-first=R.W. | chapter=Some reflections on the history of fluid dynamics | last=Anderson | first=J.D. | author-link=John D. Anderson | year=2016 | publisher=CRC Press | isbn=9781439849576|chapter-url={{google books|id=TQfYCwAAQBAJ|page=2-1|plainurl=yes|keywords=Anderson}} }}</ref><ref>{{citation | first1=O. | last1=Darrigol | first2=U. | last2=Frisch |title=From Newton's mechanics to Euler's equations | journal=Physica D: Nonlinear Phenomena | volume=237 | issue=14–17 | year=2008 | pages=1855–1869 | doi=10.1016/j.physd.2007.08.003 | bibcode=2008PhyD..237.1855D }}</ref> The principle is only applicable for [[isentropic flow]]s: when the effects of [[irreversible process]]es (like [[turbulence]]) and non-[[adiabatic process]]es (e.g. [[heat radiation]]) are small and can be neglected.

Bernoulli's principle can be applied to various types of fluid flow, resulting in various forms of ''Bernoulli's equation''; there are different forms of Bernoulli's equation for different types of flow. The simple form of Bernoulli's equation is valid for [[incompressible flow]]s (e.g. most [[liquid]] flows and [[gas]]es moving at low [[Mach number]]). More advanced forms may  be applied to [[compressible flow]]s at higher [[Mach number]]s (see [[Bernoulli's equation#Derivations of the Bernoulli equation|the derivations of the Bernoulli equation]]). <!-- This was previously deleted and had to be restored. Please state the criteria for the use of Bernoulli's principle. If there are none, don't just delete it, state it or preferably explain it. -->

Bernoulli's principle can be derived from the principle of [[conservation of energy]]. This states that, in a steady flow, the sum of all forms of energy in a fluid along a [[Streamlines, streaklines, and pathlines|streamline]] is the same at all points on that streamline. This requires that the sum of [[kinetic energy]], [[potential energy]] and [[internal energy]] remains constant.<ref name="Batchelor2000" />{{rp|at= §  3.5}} Thus an increase in the speed of the fluid – implying an increase in its kinetic energy ([[dynamic pressure]]) – occurs with a simultaneous decrease in (the sum of) its potential energy (including the [[static pressure]]) and internal energy. If the fluid is flowing out of a reservoir, the sum of all forms of energy is the same on all streamlines because in a reservoir the energy per unit volume (the sum of pressure and [[gravitational potential]] {{mvar|ρ&thinsp;g&thinsp;h}}) is the same everywhere.<ref name="Streeter1966">{{cite book|last=Streeter|first=Victor Lyle |title=Fluid mechanics|url=https://books.google.com/books?id=G6RRAAAAMAAJ|year=1966|publisher=McGraw-Hill|location=New York}}</ref>{{rp|at=Example 3.5}}

Bernoulli's principle can also be derived directly from [[Isaac Newton]]'s [[Newton's laws of motion#Newton.27s second law|Second Law of Motion]].  If a small volume of fluid is flowing horizontally from a region of high pressure to a region of low pressure, then there is more pressure behind than in front. This gives a net force on the volume, accelerating it along the streamline.{{efn|If the particle is in a region of varying pressure (a non-vanishing pressure gradient in the {{mvar|x}}-direction) and if the particle has a finite size {{mvar|l}}, then the front of the particle will be ‘seeing’ a different pressure from the rear. More precisely, if the pressure drops in the {{mvar|x}}-direction ({{math|{{sfrac|d''p''|d''x''}} < 0}}) the pressure at the rear is higher than at the front and the particle experiences a (positive) net force. According to Newton’s second law, this force causes an acceleration and the particle’s velocity increases as it moves along the streamline... Bernoulli's equation describes this mathematically (see the complete derivation in the appendix).<ref name="Babinsky2003">{{citation
  | journal=Physics Education
  | first=Holger
  | last=Babinsky
  |date=November 2003
  | url=http://www.iop.org/EJ/article/0031-9120/38/6/001/pe3_6_001.pdf
  | title=How do wings work?
  | doi=10.1088/0031-9120/38/6/001
  | bibcode=2003PhyEd..38..497B
  | volume=38
  | issue=6
  | pages=497–503
  }}</ref> }}{{efn|Acceleration of air is caused by pressure gradients. Air is accelerated in direction of the velocity if the pressure goes down. Thus the decrease of pressure is the cause of a higher velocity.<ref name="Weltner">"{{Citation|last1=Weltner |first1=Klaus |last2=Ingelman-Sundberg |first2=Martin |title=Misinterpretations of Bernoulli's Law |url=http://user.uni-frankfurt.de/~weltner/Mis6/mis6.html |url-status=dead |archiveurl=https://web.archive.org/web/20090429040229/http://user.uni-frankfurt.de/~weltner/Mis6/mis6.html |archivedate=April 29, 2009 }}</ref>}}{{efn|The idea is that as the parcel moves along, following a streamline, as it moves into an area of higher pressure there will be higher pressure ahead (higher than the pressure behind) and this will exert a force on the parcel, slowing it down. Conversely if the parcel is moving into a region of lower pressure, there will be a higher pressure behind it (higher than the pressure ahead), speeding it up. As always, any unbalanced force will cause a change in momentum (and velocity), as required by Newton’s laws of motion.<ref name="Denker2005">{{Cite web | title = 3 Airfoils and Airflow | last = Denker | first = John S. | work = See How It Flies | date = 2005 | accessdate = 2018-07-27 | url = http://www.av8n.com/how/htm/airfoils.html | quote =  }}</ref>
}}

Fluid particles are subject only to pressure and their own weight. If a fluid is flowing horizontally and along a section of a streamline, where the speed increases it can only be because the fluid on that section has moved from a region of higher pressure to a region of lower pressure; and if its speed decreases, it can only be because it has moved from a region of lower pressure to a region of higher pressure. Consequently, within a fluid flowing horizontally, the highest speed occurs where the pressure is lowest, and the lowest speed occurs where the pressure is highest.<ref>Resnick, R. and Halliday, D. (1960), section 18-4, ''Physics'', John Wiley & Sons, Inc.</ref>

== Incompressible flow equation ==<!-- [[total pressure]] and [[energy head]] redirect to here -->
In most flows of liquids, and of gases at low [[Mach number]], the [[density]] of a fluid parcel can be considered to be constant, regardless of pressure variations in the flow.  Therefore, the fluid can be considered to be incompressible and these flows are called incompressible flows. Bernoulli performed his experiments on liquids, so his equation in its original form is valid only for incompressible flow.
A common form of Bernoulli's equation, valid at any arbitrary point along a [[Streamlines, streaklines, and pathlines|streamline]], is:
{{NumBlk|:|<math>\frac{v^2}{2} + gz + \frac{p}{\rho} = \text{constant}</math>|{{EquationRef|A}}}}

where:
:{{mvar|v}} is the fluid flow [[speed]] at a point on a streamline,
:{{mvar|g}} is the [[Earth's gravity|acceleration due to gravity]],
:{{mvar|z}} is the [[elevation]] of the point above a reference plane, with the positive {{mvar|z}}-direction pointing upward – so in the direction opposite to the gravitational acceleration,
:{{mvar|p}} is the [[pressure]] at the chosen point, and
:{{mvar|ρ}} is the [[density]] of the fluid at all points in the fluid.

The constant on the right-hand side of the equation depends only on the streamline chosen, whereas {{mvar|v}}, {{mvar|z}} and {{mvar|p}} depend on the particular point on that streamline.

The following assumptions must be met for this Bernoulli equation to apply:<ref name="Batchelor2000" />{{rp|page=265}}
* the flow must be [[Fluid dynamics|steady]], i.e. the flow parameters (velocity, density, etc...) at any point cannot change with time,
* the flow must be incompressible – even though pressure varies, the density must remain constant along a streamline;
* friction by [[Viscosity|viscous]] forces must be negligible.

For [[conservative force]] fields (not limited to the gravitational field), Bernoulli's equation can be generalized as:<ref name="Batchelor2000" />{{rp|page=265}}

:<math>\frac{v^2}{2} + \Psi + \frac{p}{\rho} = \text{constant}</math>

where {{mvar|Ψ}} is the [[conservative force|force potential]] at the point considered on the streamline. E.g. for the Earth's gravity {{math|''Ψ'' {{=}} ''gz''}}.

By multiplying with the fluid density {{mvar|ρ}}, equation ({{EquationNote|A}}) can be rewritten as:

:<math>\tfrac12 \rho v^2 + \rho g z + p = \text{constant}</math>

or:

:<math>q + \rho g h = p_0 + \rho g z = \text{constant}</math>

where
*{{math|''q'' {{=}} {{sfrac|1|2}}''ρv''<sup>2</sup>}} is [[dynamic pressure]],
*{{math|''h'' {{=}} ''z'' + {{sfrac|''p''|''ρg''}}}} is the [[piezometric head]] or [[hydraulic head]] (the sum of the elevation {{mvar|z}} and the [[pressure head]])<ref name="Mulley2004">{{cite book|last=Mulley|first=Raymond |title=Flow of Industrial Fluids: Theory and Equations|url=https://books.google.com/?id=LuUZ0OG6EZMC|year=2004|publisher=CRC Press|isbn=978-0-8493-2767-4|pages=43–44}}</ref><ref name="Chanson2004">{{cite book|last=Chanson|first=Hubert |authorlink=Hubert Chanson|title=Hydraulics of Open Channel Flow|url={{google books|id=VCNmKQI6GiEC|plainurl=yes|page=22|keywords=hydraulic head}}|year=2004|publisher=Elsevier|isbn=978-0-08-047297-3|page=22}}</ref> and
*{{math|''p''<sub>0</sub> {{=}} ''p'' + ''q''}} is the [[total pressure]] (the sum of the static pressure {{math|p}} and dynamic pressure {{math|q}}).<ref name="Oerteletal2004">{{Cite book | title=Prandtl's Essentials of Fluid Mechanics | first1=Herbert | last1=Oertel | first2=Ludwig | last2= Prandtl | first3=M. | last3=Böhle | first4=Katherine | last4=Mayes | publisher=Springer | year=2004 | isbn=978-0-387-40437-0 | pages=70–71|url=https://books.google.com/books?id=J2NBBJDO79MC&pg=PA70 }}</ref>

The constant in the Bernoulli equation can be normalised. A common approach is in terms of '''total head''' or '''energy head''' {{mvar|H}}:

:<math>H = z + \frac{p}{\rho g} + \frac{v^2}{2g} = h + \frac{v^2}{2g},</math>

The above equations suggest there is a flow speed at which pressure is zero, and at even higher speeds the pressure is negative. Most often, gases and liquids are not capable of negative absolute pressure, or even zero pressure, so clearly Bernoulli's equation ceases to be valid before zero pressure is reached. In liquids – when the pressure becomes too low – [[cavitation]] occurs. The above equations use a linear relationship between flow speed squared and pressure. At higher flow speeds in gases, or for [[sound]] waves in liquid, the changes in mass density become significant so that the assumption of constant density is invalid.

=== Simplified form ===
In many applications of Bernoulli's equation, the change in the {{mvar|ρgz}} term along the streamline is so small compared with the other terms that it can be ignored. For example, in the case of aircraft in flight, the change in height {{mvar|z}} along a streamline is so small the {{mvar|ρgz}} term can be omitted.  This allows the above equation to be presented in the following simplified form:

:<math>p + q = p_0</math>

where {{math|''p''<sub>0</sub>}} is called "total pressure", and {{mvar|q}} is "[[dynamic pressure]]".<ref>{{cite web|title = Bernoulli's Equation| publisher = NASA Glenn Research Center| url =http://www.grc.nasa.gov/WWW/K-12/airplane/bern.htm|accessdate = 2009-03-04 }}</ref>  Many authors refer to the [[pressure]] {{mvar|p}} as [[static pressure]] to distinguish it from total pressure {{math|''p''<sub>0</sub>}} and [[dynamic pressure]] {{mvar|q}}. In ''Aerodynamics'', L.J. Clancy writes: "To distinguish it from the total and dynamic pressures, the actual pressure of the fluid, which is associated not with its motion but with its state, is often referred to as the static pressure, but where the term pressure alone is used it refers to this static pressure."<ref name="Clancy1975" />{{rp|at= § 3.5}}

The simplified form of Bernoulli's equation can be summarized in the following memorable word equation:<ref name="Clancy1975" />{{rp|at= § 3.5}}
:static pressure + dynamic pressure = total pressure

Every point in a steadily flowing fluid, regardless of the fluid speed at that point, has its own unique static pressure {{mvar|p}} and dynamic pressure {{mvar|q}}. Their sum {{math|''p'' + ''q''}} is defined to be the total pressure {{math|''p''<sub>0</sub>}}. The significance of Bernoulli's principle can now be summarized as "total pressure is constant along a streamline".

If the fluid flow is [[irrotational flow|irrotational]], the total pressure on every streamline is the same and Bernoulli's principle can be summarized as "total pressure is constant everywhere in the fluid flow".<ref name="Clancy1975" />{{rp|at=Equation 3.12}} It is reasonable to assume that irrotational flow exists in any situation where a large body of fluid is flowing past a solid body.  Examples are aircraft in flight, and ships moving in open bodies of water.  However, it is important to remember that Bernoulli's principle does not apply in the [[boundary layer]] or in fluid flow through long [[Pipe flow|pipes]].

If the fluid flow at some point along a streamline is brought to rest, this point is called a stagnation point, and at this point the total pressure is equal to the [[stagnation pressure]].

=== Applicability of incompressible flow equation to flow of gases ===
Bernoulli's equation is sometimes valid for the flow of gases: provided that there is no transfer of kinetic or potential energy from the gas flow to the compression or expansion of the gas.  If both the gas pressure and volume change simultaneously, then work will be done on or by the gas.  In this case, Bernoulli's equation – in its incompressible flow form – cannot be assumed to be valid.  However, if the gas process is entirely [[isobaric process|isobaric]], or [[isochoric process|isochoric]], then no work is done on or by the gas, (so the simple energy balance is not upset).  According to the gas law, an isobaric or isochoric process is ordinarily the only way to ensure constant density in a gas.  Also the gas density will be proportional to the ratio of pressure and absolute [[temperature]], however this ratio will vary upon compression or expansion, no matter what non-zero quantity of heat is added or removed.  The only exception is if the net heat transfer is zero, as in a complete thermodynamic cycle, or in an individual [[isentropic]] ([[friction]]less [[adiabatic]]) process, and even then this reversible process must be reversed, to restore the gas to the original pressure and specific volume, and thus density.  Only then is the original, unmodified Bernoulli equation applicable.  In this case the equation can be used if the flow speed of the gas is sufficiently below the [[speed of sound]], such that the variation in density of the gas (due to this effect) along each [[Streamlines, streaklines and pathlines|streamline]] can be ignored.  Adiabatic flow at less than Mach 0.3 is generally considered to be slow enough.

=== Unsteady potential flow ===

The Bernoulli equation for unsteady potential flow is used in the theory of [[ocean surface wave]]s and [[acoustics]].

For an [[irrotational flow]], the [[flow velocity]] can be described as the [[gradient]] {{math|∇''φ''}} of a [[velocity potential]] {{mvar|φ}}. In that case, and for a constant [[density]] {{mvar|ρ}}, the [[momentum]] equations of the [[Euler equations (fluid dynamics)|Euler equations]] can be integrated to:<ref name="Batchelor2000" />{{rp|page=383}}
:<math>\frac{\partial \varphi}{\partial t} + \tfrac12 v^2 + \frac{p}{\rho} + gz = f(t),</math>

which is a Bernoulli equation valid also for unsteady—or time dependent—flows. Here {{math|{{sfrac|∂''φ''|∂''t''}}}} denotes the [[partial derivative]] of the velocity potential {{mvar|φ}} with respect to time {{mvar|t}}, and {{math|''v'' {{=}} {{abs|∇''φ''}}}} is the flow speed.
The function {{math|''f''(''t'')}} depends only on time and not on position in the fluid. As a result, the Bernoulli equation at some moment {{mvar|t}} does not only apply along a certain streamline, but in the whole fluid domain. This is also true for the special case of a steady irrotational flow, in which case {{mvar|f}} and {{math|∂''φ''/∂''t''}} are constants so equation ({{EquationNote|A}}) can be applied in every point of the fluid domain.<ref name="Batchelor2000" />{{rp|page=383}}

Further {{math|''f''(''t'')}} can be made equal to zero by incorporating it into the velocity potential using the transformation

:<math>\Phi=\varphi-\int_{t_0}^t f(\tau)\, \mathrm{d}\tau,</math>

resulting in

:<math>\frac{\partial \Phi}{\partial t} + \tfrac12 v^2 + \frac{p}{\rho} + gz=0.</math>

Note that the relation of the potential to the flow velocity is unaffected by this transformation: {{math|∇''Φ'' {{=}} ∇''φ''}}.

The Bernoulli equation for unsteady potential flow also appears to play a central role in [[Luke's variational principle]], a variational description of free-surface flows using the [[Lagrangian mechanics|Lagrangian]] (not to be confused with [[Lagrangian coordinates]]).

== Compressible flow equation ==
Bernoulli developed his principle from his observations on liquids, and his equation is applicable only to incompressible fluids, and steady compressible fluids up to approximately [[Mach number]] 0.3.<ref>White, Frank M. ''Fluid Mechanics'', 6th ed. McGraw-Hill International Edition. p. 602.</ref> It is possible to use the fundamental principles of physics to develop similar equations applicable to compressible fluids. There are numerous equations, each tailored for a particular application, but all are analogous to Bernoulli's equation and all rely on nothing more than the fundamental principles of physics such as Newton's laws of motion or the [[first law of thermodynamics]].

=== Compressible flow in fluid dynamics ===
For a compressible fluid, with a [[barotropic]] [[equation of state]], and under the action of [[conservative force]]s,<ref name="ClarkeCarswell2007">{{cite book |last1=Clarke |first1=Cathie |last2=Carswell |first2=Bob |authorlink2=Bob Carswell |title=Principles of Astrophysical Fluid Dynamics |url=https://books.google.com/books?id=a2SW7XJ89H0C&pg=PA161 |year=2007 |publisher=Cambridge University Press |isbn=978-1-139-46223-5 |page=161}}</ref>

:<math>\frac {v^2}{2}+ \int_{p_1}^p \frac {\mathrm{d}\tilde{p}}{\rho\left(\tilde{p}\right)} + \Psi = \text{constant (along a streamline)}</math>

where:
*{{mvar|p}} is the [[pressure]]
*{{mvar|ρ}} is the [[density]] and <math>\rho(\tilde{p})</math> indicates that it is a function of pressure
*{{mvar|v}} is the [[flow speed]]
*{{mvar|Ψ}} is the potential associated with the conservative force field, often the [[gravitational potential]]

In engineering situations, elevations are generally small compared to the size of the Earth, and the time scales of fluid flow are small enough to consider the equation of state as [[adiabatic]]. In this case, the above equation for an [[ideal gas]] becomes:<ref name="Clancy1975" />{{rp|at= § 3.11}}

:<math>\frac {v^2}{2}+ gz+\left(\frac {\gamma}{\gamma-1}\right)\frac {p}{\rho}   = \text{constant (along a streamline)}</math>

where, in addition to the terms listed above:
*{{mvar|γ}} is the [[Heat capacity ratio|ratio of the specific heats]] of the fluid
*{{mvar|g}} is the acceleration due to gravity
*{{mvar|z}} is the elevation of the point above a reference plane

In many applications of compressible flow, changes in elevation are negligible compared to the other terms, so the term {{mvar|gz}} can be omitted. A very useful form of the equation is then:

:<math>\frac {v^2}{2}+\left( \frac {\gamma}{\gamma-1}\right)\frac {p}{\rho}  = \left(\frac {\gamma}{\gamma-1}\right)\frac {p_0}{\rho_0}</math>

where:
*{{math|''p''<sub>0</sub>}} is the [[Stagnation pressure|total pressure]]
*{{math|''ρ''<sub>0</sub>}} is the total density

=== Compressible flow in thermodynamics ===
The most general form of the equation, suitable for use in thermodynamics in case of (quasi) steady flow, is:<ref name="Batchelor2000" />{{rp|at= § 3.5}}<ref name="LandauLifshitz1987">{{cite book |last1=Landau |first1=L.D. |author1-link=Lev Landau |last2=Lifshitz |first2=E.M. |author2-link=Evgeny Lifshitz |title=Fluid Mechanics |edition=2nd |series=[[Course of Theoretical Physics]] |publisher=Pergamon Press |year=1987 |isbn=978-0-7506-2767-2 }}</ref>{{rp|at=§ 5}}<ref name="VanWylenSonntag1965">{{cite book |last1=Van Wylen |first1=Gordon J. |authorlink1=Gordon Van Wylen |last2=Sonntag |first2=Richard E. |title=Fundamentals of Classical Thermodynamics |url=https://books.google.com/books?id=Ahx-JsR3_OgC |year=1965 |publisher=John Wiley and Sons |location=New York}}</ref>{{rp|at=§ 5.9}}

:<math>\frac{v^2}{2} + \Psi + w = \text{constant}.</math>

Here {{mvar|w}} is the [[enthalpy]] per unit mass (also known as specific enthalpy), which is also often written as {{mvar|h}} (not to be confused with "head" or "height").

Note that {{math|''w'' {{=}} ''ε'' + {{sfrac|''p''|''ρ''}}}} where {{mvar|ε}} is the [[thermodynamics|thermodynamic]] energy per unit mass, also known as the [[specific energy|specific]] [[internal energy]]. So, for constant internal energy {{mvar|ε}} the equation reduces to the incompressible-flow form.

The constant on the right-hand side is often called the Bernoulli constant, and denoted {{mvar|b}}. For steady inviscid [[adiabatic process|adiabatic]] flow with no additional sources or sinks of energy, {{mvar|b}} is constant along any given streamline. More generally, when {{mvar|b}} may vary along streamlines, it still proves a useful parameter, related to the "head" of the fluid (see below).

When the change in {{mvar|Ψ}} can be ignored, a very useful form of this equation is:

:<math>\frac{v^2}{2} + w = w_0</math>

where {{math|''w''<sub>0</sub>}} is total enthalpy. For a calorically perfect gas such as an ideal gas, the enthalpy is directly proportional to the temperature, and this leads to the concept of the total (or stagnation) temperature.

When [[shock wave]]s are present, in a [[frame of reference|reference frame]] in which the shock is stationary and the flow is steady, many of the parameters in the Bernoulli equation suffer abrupt changes in passing through the shock. The Bernoulli parameter itself, however, remains unaffected. An exception to this rule is radiative shocks, which violate the assumptions leading to the Bernoulli equation, namely the lack of additional sinks or sources of energy.

== Derivations of the Bernoulli equation ==
:{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
!Bernoulli equation for incompressible fluids
|-
|The Bernoulli equation for incompressible fluids can be derived by either [[integral|integrating]] [[Newton's second law of motion]] or by applying the law of [[conservation of energy]] between two sections along a streamline, ignoring [[viscosity]], compressibility, and thermal effects.
; Derivation through integrating Newton's Second Law of Motion
The simplest derivation is to first ignore gravity and consider constrictions and expansions in pipes that are otherwise straight, as seen in [[Venturi effect]].  Let the {{mvar|x}} axis be directed down the axis of the pipe.

Define a parcel of fluid moving through a pipe with cross-sectional area {{mvar|A}}, the length of the parcel is {{math|d''x''}}, and the volume of the parcel {{math|''A'' d''x''}}. If [[mass density]] is {{math|ρ}}, the mass of the parcel is density multiplied by its volume {{math|''m'' {{=}} ''ρA'' d''x''}}. The change in pressure over distance {{math|d''x''}} is {{math|d''p''}} and [[flow velocity]] {{math|''v'' {{=}} {{sfrac|d''x''|d''t''}}}}.

Apply [[Newton's second law of motion]] (force&nbsp;= mass&nbsp;×&nbsp;acceleration) and recognizing that the effective force on the [[fluid parcel|parcel of fluid]] is {{math|−''A'' d''p''}}. If the pressure decreases along the length of the pipe, {{math|d''p''}} is negative but the force resulting in flow is positive along the {{mvar|x}} axis.

:<math>\begin{align}
m \frac{\mathrm{d}v}{\mathrm{d}t}&= F \\
\rho A \mathrm{d}x \frac{\mathrm{d}v}{\mathrm{d}t} &= -A \mathrm{d}p \\
\rho \frac{\mathrm{d}v}{\mathrm{d}t} &= -\frac{\mathrm{d}p}{\mathrm{d}x} 
\end{align}</math>

In steady flow the velocity field is constant with respect to time, {{math|''v'' {{=}} ''v''(''x'') {{=}} ''v''(''x''(''t''))}}, so {{mvar|v}} itself is not directly a function of time {{mvar|t}}. It is only when the parcel moves through {{mvar|x}} that the cross sectional area changes: {{mvar|v}} depends on {{mvar|t}} only through the cross-sectional position {{math|''x''(''t'')}}.

:<math>\begin{align}
\frac{\mathrm{d}v}{\mathrm{d}t}&= \frac{\mathrm{d}v}{\mathrm{d}x}\frac{\mathrm{d}x}{\mathrm{d}t} \\ 
&= \frac{\mathrm{d}v}{\mathrm{d}x}v \\
&=\frac{\mathrm{d}}{\mathrm{d}x} \left( \frac{v^2}{2} \right).
\end{align}</math>

With density {{mvar|ρ}} constant, the equation of motion can be written as

:<math>\frac{\mathrm{d}}{\mathrm{d}x} \left(  \rho \frac{v^2}{2} + p \right) =0</math>

by integrating with respect to {{mvar|x}}

:<math> \frac{v^2}{2} + \frac{p}{\rho}= C</math>

where {{mvar|C}} is a constant, sometimes referred to as the Bernoulli constant. It is not a [[universal constant]], but rather a constant of a particular fluid system. The deduction is: where the speed is large, pressure is low and vice versa.

In the above derivation, no external work–energy principle is invoked. Rather, Bernoulli's principle was derived by a simple manipulation of Newton's second law.
[[File:BernoullisLawDerivationDiagram.svg|thumb|center|600px|A streamtube of fluid moving to the right. Indicated are pressure, elevation, flow speed, distance ({{mvar|s}}), and cross-sectional area. Note that in this figure elevation is denoted as {{mvar|h}}, contrary to the text where it is given by {{mvar|z}}.]]
; Derivation by using conservation of energy
Another way to derive Bernoulli's principle for an incompressible flow is by applying conservation of energy.<ref name="FeynmanLeightonSands1963V2">{{cite book |first1=R.P. |last=Feynman |authorlink1=R.P. Feynman |first2=R.B. |last2=Leighton |authorlink2=R.B. Leighton |first3=M. |last3=Sands |year=1963 |title=The Feynman Lectures on Physics |isbn=978-0-201-02116-5 |volume=Vol. 2 |title-link=The Feynman Lectures on Physics}}{{rp|at= §40–3|pp=40–6 to 40–9}}</ref> In the form of the [[work (physics)|work-energy theorem]], stating that<ref>{{cite book |last=Tipler |first=Paul |title=Physics for Scientists and Engineers: Mechanics |edition=3rd extended |publisher=W. H. Freeman |year=1991 |isbn=978-0-87901-432-2}}, p. 138.</ref>
:''the change in the kinetic energy ''{{math|''E''<sub>kin</sub>}}'' of the system equals the net work {{mvar|W}} done on the system'';
::<math>W = \Delta E_\text{kin}.</math>
Therefore,
:''the [[Mechanical work|work]] done by the [[force]]s in the fluid equals increase in [[kinetic energy]].''
The system consists of the volume of fluid, initially between the cross-sections {{math|''A''<sub>1</sub>}} and {{math|''A''<sub>2</sub>}}. In the time interval {{math|Δ''t''}} fluid elements initially at the inflow cross-section {{math|''A''<sub>1</sub>}} move over a distance {{math|''s''<sub>1</sub> {{=}} ''v''<sub>1</sub> Δ''t''}}, while at the outflow cross-section the fluid moves away from cross-section {{math|''A''<sub>2</sub>}} over a distance {{math|''s''<sub>2</sub> {{=}} ''v''<sub>2</sub> Δ''t''}}. The displaced fluid volumes at the inflow and outflow are respectively {{math|''A''<sub>1</sub>''s''<sub>1</sub>}} and {{math|''A''<sub>2</sub>''s''<sub>2</sub>}}. The associated displaced fluid masses are – when {{mvar|ρ}} is the fluid's [[density|mass density]] – equal to density times volume, so {{math|''ρA''<sub>1</sub>''s''<sub>1</sub>}} and {{math|''ρA''<sub>2</sub>''s''<sub>2</sub>}}. By mass conservation, these two masses displaced in the time interval {{math|Δ''t''}} have to be equal, and this displaced mass is denoted by&nbsp;{{math|Δ''m''}}:

:<math>\begin{align}
\rho A_1 s_1 &= \rho A_1 v_1 \Delta t = \Delta m, \\
\rho A_2 s_2 &= \rho A_2 v_2 \Delta t = \Delta m.
\end{align}</math>

The work done by the forces consists of two parts:
* The ''work done by the pressure'' acting on the areas {{math|''A''<sub>1</sub>}} and {{math|''A''<sub>2</sub>}}
*: <math>W_\text{pressure}=F_{1,\text{pressure}} s_{1} - F_{2,\text{pressure}} s_2 =p_1 A_1 s_1 - p_2 A_2 s_2 = \Delta m \frac{p_1}{\rho} - \Delta m \frac{p_2}{\rho}.</math>
* The ''work done by gravity'': the gravitational potential energy in the volume {{math|''A''<sub>1</sub>''s''<sub>1</sub>}} is lost, and at the outflow in the volume {{math|''A''<sub>2</sub>''s''<sub>2</sub>}} is gained.  So, the change in gravitational potential energy {{math|Δ''E''<sub>pot,gravity</sub>}} in the time interval {{math|Δ''t''}} is
*: <math>\Delta E_\text{pot,gravity} = \Delta m\, g z_2 - \Delta m\, g z_1. </math>
:Now, the [[Energy#Potential energy|work by the force of gravity is opposite to the change in potential energy]], {{math|''W''<sub>gravity</sub> {{=}} −''ΔE''<sub>pot,gravity</sub>}}: while the force of gravity is in the negative {{mvar|z}}-direction, the work—gravity force times change in elevation—will be negative for a positive elevation change {{math|Δ''z'' {{=}} ''z''<sub>2</sub> − ''z''<sub>1</sub>}}, while the corresponding potential energy change is positive.<ref name="FeynmanLeightonSands1963V1">{{Cite book| first1=R.P. | last=Feynman |authorlink1=R. P. Feynman | first2=R.B. | last2=Leighton| authorlink2=R. B. Leighton | first3=M. | last3=Sands | year=1963 | title=The Feynman Lectures on Physics | isbn=978-0-201-02116-5|volume= Vol. 1| title-link=The Feynman Lectures on Physics }}</ref>{{rp|at= §14–3| p=14–4}} So:
::<math>W_\text{gravity} = -\Delta E_\text{pot,gravity} = \Delta m\, g z_1 - \Delta m\, g z_2.</math>
And therefore the total work done in this time interval {{math|Δ''t''}} is
:<math>W = W_\text{pressure} + W_\text{gravity}.</math>
The ''increase in kinetic energy'' is
: <math>\Delta E_\text{kin} = \tfrac12 \Delta m\, v_2^2-\tfrac12 \Delta m\, v_1^2.</math>
Putting these together, the work-kinetic energy theorem {{math|''W'' {{=}} Δ''E''<sub>kin</sub>}} gives:<ref name="FeynmanLeightonSands1963V2" />
: <math>\Delta m \frac{p_1}{\rho} - \Delta m \frac{p_2}{\rho} + \Delta m\, g z_1 - \Delta m\, g z_2 = \tfrac12 \Delta m\, v_2^2 - \tfrac12 \Delta m\, v_1^2</math>
or
: <math>\tfrac12 \Delta m\, v_1^2 + \Delta m\, g  z_1 + \Delta m \frac{p_1}{\rho} = \tfrac12 \Delta m\, v_2^2 + \Delta m\, g z_2 + \Delta m \frac{p_2}{\rho}.</math>
After dividing by the mass {{math|Δ''m'' {{=}} ''ρA''<sub>1</sub>''v''<sub>1</sub> Δ''t'' {{=}} ''ρA''<sub>2</sub>''v''<sub>2</sub> Δ''t''}} the result is:<ref name="FeynmanLeightonSands1963V2" />
: <math>\tfrac12 v_1^2 +g z_1 + \frac{p_1}{\rho}=\tfrac12 v_2^2 +g z_2 + \frac{p_2}{\rho}</math>
or, as stated in the first paragraph:
:<math>\frac{v^2}{2}+g z+\frac{p}{\rho}=C</math> {{pad|3em}} '''(Eqn. 1)''', Which is also Equation (A)
Further division by {{mvar|g}} produces the following equation. Note that each term can be described in the [[length]] dimension (such as meters). This is the head equation derived from Bernoulli's principle:
:<math>\frac{v^2}{2 g}+z+\frac{p}{\rho g}=C</math> {{pad|3em}} '''(Eqn. 2a)'''
The middle term, {{mvar|z}}, represents the potential energy of the fluid due to its elevation with respect to a reference plane. Now, {{math|z}} is called the elevation head and given the designation {{math|''z''<sub>elevation</sub>}}.

A [[free fall]]ing mass from an elevation {{math|''z'' > 0}} (in a [[vacuum]]) will reach a [[speed]]
:<math>v=\sqrt{{2 g}{z}},</math>
when arriving at elevation {{math|''z'' {{=}} 0}}. Or when we rearrange it as a ''head'':
:<math>h_v =\frac{v^2}{2 g}</math>
The [[term (mathematics)|term]] {{math|{{sfrac|''v''<sup>2</sup>|2''g''}}}} is called the ''velocity [[Hydraulic head|head]]'', expressed as a length measurement. It represents the internal energy of the fluid due to its motion.

The [[hydrostatic pressure]] ''p'' is defined as
:<math>p=p_0-\rho g z ,</math>
with {{math|''p''<sub>0</sub>}} some reference pressure, or when we rearrange it as a ''head'':
:<math>\psi=\frac{p}{\rho g}.</math>
The term {{math|{{sfrac|''p''|''ρg''}}}} is also called the ''[[pressure head]]'', expressed as a length measurement. It represents the internal energy of the fluid due to the pressure exerted on the container. When we combine the head due to the flow speed and the head due to static pressure with the elevation above a reference plane, we obtain a simple relationship useful for incompressible fluids using the velocity head, elevation head, and pressure head.
:<math>h_{v} + z_\text{elevation} + \psi = C</math> {{pad|3em}} '''(Eqn. 2b)'''
If we were to multiply Eqn. 1 by the density of the fluid, we would get an equation with three pressure terms:
:<math>\frac{\rho v^2}{2}+ \rho g z + p=C</math> {{pad|3em}} '''(Eqn. 3)'''
We note that the pressure of the system is constant in this form of the Bernoulli equation. If the static pressure of the system (the far right term) increases, and if the pressure due to elevation (the middle term) is constant, then we know that the dynamic pressure (the left term) must have decreased. In other words, if the speed of a fluid decreases and it is not due to an elevation difference, we know it must be due to an increase in the static pressure that is resisting the flow.

All three equations are merely simplified versions of an energy balance on a system.
|}
:{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
!Bernoulli equation for compressible fluids
|-
|The derivation for compressible fluids is similar. Again, the derivation depends upon (1) conservation of mass, and (2) conservation of energy. Conservation of mass implies that in the above figure, in the interval of time {{math|Δ''t''}}, the amount of mass passing through the boundary defined by the area {{math|''A''<sub>1</sub>}} is equal to the amount of mass passing outwards through the boundary defined by the area {{math|''A''<sub>2</sub>}}:
:<math>0= \Delta M_1 - \Delta M_2 = \rho_1 A_1 v_1 \, \Delta t - \rho_2 A_2 v_2 \, \Delta t</math>.
Conservation of energy is applied in a similar manner: It is assumed that the change in energy of the volume
of the streamtube bounded by {{math|''A''<sub>1</sub>}} and {{math|''A''<sub>2</sub>}} is due entirely to energy entering or leaving through one or the other of these two boundaries. Clearly, in a more complicated situation such as a fluid flow coupled with radiation, such conditions are not met. Nevertheless, assuming this to be the case and assuming the flow is steady so that the net change in the energy is zero,
:<math>0= \Delta E_1 - \Delta E_2 </math>
where {{math|Δ''E''<sub>1</sub>}} and {{math|Δ''E''<sub>2</sub>}} are the energy entering through {{math|''A''<sub>1</sub>}} and leaving through {{math|''A''<sub>2</sub>}}, respectively. The energy entering through {{math|''A''<sub>1</sub>}} is the sum of the kinetic energy entering, the energy entering in the form of potential gravitational energy of the fluid, the fluid thermodynamic internal energy per unit of mass ({{math|''ε''<sub>1</sub>}}) entering, and the energy entering in the form of mechanical {{math|''p'' d''V''}} work:
:<math>\Delta E_1 = \left(\tfrac12 \rho_1 v_1^2 + \Psi_1 \rho_1 + \epsilon_1 \rho_1  + p_1 \right) A_1 v_1 \, \Delta t</math>
where {{math|''Ψ'' {{=}} ''gz''}} is a [[conservative force|force potential]] due to the [[Earth's gravity]], {{mvar|g}} is acceleration due to gravity, and {{mvar|z}} is elevation above a reference plane. A similar expression for {{math|Δ''E''<sub>2</sub>}} may easily be constructed.
So now setting {{math|0 {{=}} Δ''E''<sub>1</sub> − Δ''E''<sub>2</sub>}}:
:<math>0 = \left(\tfrac12 \rho_1 v_1^2+ \Psi_1 \rho_1 + \epsilon_1 \rho_1  + p_1 \right) A_1 v_1 \, \Delta t - \left(\tfrac12 \rho_2 v_2^2 + \Psi_2 \rho_2 + \epsilon_2 \rho_2  + p_2 \right) A_2 v_2 \, \Delta t</math>
which can be rewritten as:
:<math> 0 = \left(\tfrac12 v_1^2 + \Psi_1 + \epsilon_1  + \frac{p_1}{\rho_1} \right) \rho_1 A_1 v_1 \, \Delta t - \left(\tfrac12 v_2^2  + \Psi_2 + \epsilon_2  + \frac{p_2}{\rho_2} \right) \rho_2 A_2 v_2 \, \Delta t </math>
Now, using the previously-obtained result from conservation of mass, this may be simplified to obtain
:<math> \tfrac12 v^2 + \Psi + \epsilon + \frac{p}{\rho} = \text{constant} \equiv b </math>
which is the Bernoulli equation for compressible flow.

An equivalent expression can be written in terms of fluid enthalpy ({{mvar|h}}):

:<math> \tfrac12 v^2 + \Psi + h = \text{constant} \equiv b </math>

|}

== Applications ==
[[File:Cloud over A340 wing.JPG|thumb|right|Condensation visible over the upper surface of an [[Airbus A340]] wing caused by the fall in temperature [[Gay-Lussac's law#Pressure-temperature law|accompanying]] the fall in pressure.]]
In modern everyday life there are many observations that can be successfully explained by application of Bernoulli's principle, even though no real fluid is entirely inviscid<ref name="Thomas2010">{{cite journal|journal=Physics Today|date= May 2010|title=The Nearly Perfect Fermi Gas|first= John E.|last= Thomas|p =34|url=https://physics.ncsu.edu/jet/publications/pdf/Physicstoday2010May.pdf}}</ref> and a small viscosity often has a large effect on the flow.

<!-- Please do not edit this entry regarding lift unless you have read the Talk Page -->
*Bernoulli's principle can be used to calculate the lift force on an airfoil, if the behaviour of the fluid flow in the vicinity of the foil is known. For example, if the air flowing past the top surface of an aircraft wing is moving faster than the air flowing past the bottom surface, then Bernoulli's principle implies that the [[pressure]] on the surfaces of the wing will be lower above than below. This pressure difference results in an upwards [[lift (force)|lifting force]].{{efn|"When a stream of air flows past an airfoil, there are local changes in velocity round the airfoil, and consequently changes in static pressure, in accordance with Bernoulli's Theorem. The distribution of pressure determines the lift, pitching moment and form drag of the airfoil, and the position of its centre of pressure."<ref name="Clancy1975" />{{rp|at= § 5.5}} }}<ref>Resnick, R. and Halliday, D. (1960), ''Physics'', Section&nbsp;18–5, John Wiley &amp; Sons, Inc., New York ("[[Streamlines, streaklines, and pathlines|Streamlines]] are closer together above the wing than they are below so that Bernoulli's principle predicts the observed upward dynamic lift.")</ref> Whenever the distribution of speed past the top and bottom surfaces of a wing is known, the lift forces can be calculated (to a good approximation) using Bernoulli's equations<ref name="Eastlake">{{Cite journal
 | last1=Eastlake
 | first1=Charles N.
 | title=An Aerodynamicist's View of Lift, Bernoulli, and Newton
 | url=http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/Bernoulli_Newton_lift.pdf
 | journal=The Physics Teacher
 | volume=40
 | issue=3
 | pages=166–173
 | date=March 2002
 | doi=10.1119/1.1466553
| bibcode=2002PhTea..40..166E
 }} "The resultant force is determined by integrating the surface-pressure
distribution over the surface area of the airfoil."</ref> – established by Bernoulli over a century before the first man-made wings were used for the purpose of flight. Bernoulli's principle does not explain why the air flows faster past the top of the wing and slower past the underside. See the article on [[Lift (force)|aerodynamic lift]] for more info.

*The [[carburettor]] used in many reciprocating engines contains a [[Venturi effect|venturi]] to create a region of low pressure to draw fuel into the carburettor and mix it thoroughly with the incoming air.  The low pressure in the throat of a venturi can be explained by Bernoulli's principle; in the narrow throat, the air is moving at its fastest speed and therefore it is at its lowest pressure.
*An [[injector]] on a steam locomotive (or static boiler). 
*The [[pitot tube]] and [[Pitot-static system|static port]] on an aircraft are used to determine the [[airspeed]] of the aircraft. These two devices are connected to the [[airspeed indicator]], which determines the [[dynamic pressure]] of the airflow past the aircraft.  Dynamic pressure is the difference between [[stagnation pressure]] and [[static pressure]].  Bernoulli's principle is used to calibrate the airspeed indicator so that it displays the [[indicated airspeed]] appropriate to the dynamic pressure.<ref name="Clancy1975" />{{rp|at= § 3.8}}
*A [[De Laval nozzle]] utilizes Bernoulli's principle to create a [[force]] by turning pressure energy generated by the combustion of [[Rocket propellant|propellants]] into velocity. This then generates thrust by way of [[Newton's laws of motion|Newton's third law of motion]].
*The flow speed of a fluid can be measured using a device such as a [[Venturi meter]] or an [[orifice plate]], which can be placed into a pipeline to reduce the diameter of the flow. For a horizontal device, the [[Continuity equation#Fluid dynamics|continuity equation]] shows that for an incompressible fluid, the reduction in diameter will cause an increase in the fluid flow speed. Subsequently, Bernoulli's principle then shows that there must be a decrease in the pressure in the reduced diameter region. This phenomenon is known as the [[Venturi effect]].
*The maximum possible drain rate for a tank with a hole or tap at the base can be calculated directly from Bernoulli's equation, and is found to be proportional to the square root of the height of the fluid in the tank.  This is [[Torricelli's law]], showing that Torricelli's law is compatible with Bernoulli's principle.  [[Viscosity]] lowers this drain rate.  This is reflected in the discharge coefficient, which is a function of the Reynolds number and the shape of the orifice.<ref>''Mechanical Engineering Reference Manual'' Ninth Edition</ref>
<!-- SOURCE additions here -->

*The [[Bernoulli grip]] relies on this principle to create a non-contact adhesive force between a surface and the gripper.
*Bernoulli's principle is also applicable in the swinging of a cricket ball. During a cricket match, bowlers continually polish one side of the ball. After some time, one side is quite rough and the other is still smooth. Hence, when the ball is bowled and passes through air, the speed on one side of the ball is faster than on the other, due to this difference in smoothness, and this results in a pressure difference between the sides; this leads to the ball rotating ("swinging") while travelling through the air, giving advantage to the bowlers.

== Misunderstandings about the generation of lift ==
{{main|Lift (force)}}
Many explanations for the generation of lift (on [[airfoil]]s, [[Propeller (aircraft)|propeller]] blades, etc.) can be found; some of these explanations can be misleading, and some are false.<ref name=NASA_Incorrect_Theory1>{{cite web |url=http://www.grc.nasa.gov/WWW/K-12/airplane/wrong1.html |date=2006-03-15 |title=Incorrect Lift Theory |author=Glenn Research Center |publisher=NASA |accessdate=2010-08-12}}</ref> There has been debate about whether lift is best introduced to students using Bernoulli's principle or [[Newton's laws of motion]]. Modern writings agree that both Bernoulli's principle and Newton's laws are relevant, and either can be used to correctly describe lift.<ref name="Chanson2004" /><ref>{{cite web |url=http://www.grc.nasa.gov/WWW/K-12/airplane/bernnew.html |title=Newton vs Bernoulli}}</ref><ref name="Ison2006">{{cite web |title=Bernoulli Or Newton: Who's Right About Lift? |last=Ison |first=David |work=Plane & Pilot Magazine |date=1 July 2006 |accessdate=2018-07-27 |url=https://www.planeandpilotmag.com/article/bernoulli-or-newton-whos-right-about-lift/#.W1sKrtIza00 |quote= }}</ref>

Several of these explanations use the Bernoulli principle to connect the flow kinematics to the flow-induced pressures. In cases of [[Equal transit-time fallacy|incorrect (or partially correct) explanations relying on the Bernoulli principle]], the errors generally occur in the assumptions on the flow kinematics and how these are produced. It is not the Bernoulli principle itself that is questioned, because this principle is well established (the airflow above the wing ''is'' faster, the question is ''why'' it is faster).<ref name=Phillips>{{cite book |first=O.M. |last=Phillips |author-link=Owen Martin Phillips |title=The dynamics of the upper ocean |publisher=Cambridge University Press |year=1977 |edition=2nd |isbn=978-0-521-29801-8}} Section 2.4.</ref><ref name="Batchelor2000" />{{rp|at=Section 3.5 and 5.1}}<ref name="Lamb1994">{{cite book |first=H. |last=Lamb |authorlink=Horace Lamb |year=1993 |title=Hydrodynamics |publisher=Cambridge University Press |edition=6th |isbn=978-0-521-45868-9 |orig-year=1879}}</ref>{{rp|at=§17–§29}}<ref name="Weltner_Physics_of_Flight_Reviewed">{{cite web |last1=Weltner |first1=Klaus |last2=Ingelman-Sundberg |first2=Martin |title=Physics of Flight – reviewed |url=http://www.angelfire.com/dc/nova/flight/PHYSIC4.html}} "The conventional explanation of aerodynamical lift based on Bernoulli’s law and velocity differences mixes up ''cause'' and ''effect''. The faster flow at the upper side of the wing is the consequence of low pressure and not its cause."</ref>

== Misapplications of Bernoulli's principle in common classroom demonstrations ==
There are several common classroom demonstrations that are sometimes incorrectly explained using Bernoulli's principle.<ref>"Bernoulli's law and experiments attributed to it are fascinating. Unfortunately some of these experiments are explained erroneously..."  {{cite web |title=Misinterpretations of Bernoulli's Law |last1=Weltner |first1=Klaus |last2=Ingelman-Sundberg |first2=Martin |publisher=Department of Physics, University Frankfurt |url=http://www-stud.rbi.informatik.uni-frankfurt.de/~plass/MIS/mis6.html |accessdate=June 25, 2012 |url-status=dead |archiveurl=https://web.archive.org/web/20120621073812/http://www-stud.rbi.informatik.uni-frankfurt.de/~plass/MIS/mis6.html |archivedate=June 21, 2012 }}</ref> One involves holding a piece of paper horizontally so that it droops downward and then blowing over the top of it.  As the demonstrator blows over the paper, the paper rises. It is then asserted that this is because "faster moving air has lower pressure".<ref>"This occurs because of Bernoulli’s principle — fast-moving air has lower pressure than non-moving air." Make Magazine http://makeprojects.com/Project/Origami-Flying-Disk/327/1</ref><ref name=Minnesota>" Faster-moving fluid, lower pressure. ... When the demonstrator holds the paper in front of his mouth and blows across the top, he is creating an area of faster-moving air." University of Minnesota School of Physics and Astronomy http://www.physics.umn.edu/outreach/pforce/circus/Bernoulli.html {{Webarchive|url=https://web.archive.org/web/20120310190206/http://www.physics.umn.edu/outreach/pforce/circus/Bernoulli.html |date=2012-03-10 }}</ref><ref>"Bernoulli's Principle states that faster moving air has lower pressure... You can demonstrate Bernoulli's Principle by blowing over a piece of paper held horizontally across your lips." {{cite web |publisher=Tall Ships Festival – Channel Islands Harbor |title=Educational Packet |url=http://www.tallshipschannelislands.com/PDFs/Educational_Packet.pdf |accessdate=June 25, 2012 |url-status=dead |archiveurl=https://web.archive.org/web/20131203014334/http://www.tallshipschannelislands.com/PDFs/Educational_Packet.pdf |archivedate=December 3, 2013 }}</ref>

One problem with this explanation can be seen by blowing along the bottom of the paper: were the deflection due simply to faster moving air one would expect the paper to deflect downward, but the paper deflects upward regardless of whether the faster moving air is on the top or the bottom.<ref>"If the lift in figure A were caused by "Bernoulli's principle," then the paper in figure B should droop further when air is blown beneath it. However, as shown, it raises when the upward pressure gradient in downward-curving flow adds to atmospheric pressure at the paper lower surface." {{cite web |first=Gale M. |last=Craig |title=Physical Principles of Winged Flight |url=http://www.rcgroups.com/forums/showatt.php?attachmentid=5305482 |accessdate=March 31, 2016 }}</ref> Another problem is that when the air leaves the demonstrator's mouth it has the ''same'' pressure as the surrounding air;<ref>"In fact, the pressure in the air blown out of the lungs is equal to that of the surrounding air..." Babinsky  http://iopscience.iop.org/0031-9120/38/6/001/pdf/pe3_6_001.pdf </ref> the air does not have lower pressure just because it is moving; in the demonstration, the static pressure of the air leaving the demonstrator's mouth is ''equal'' to the pressure of the surrounding air.<ref>{{cite journal|quote=...air does not have a reduced lateral pressure (or static pressure...) simply because it is caused to move, the static pressure of free air does not decrease as the speed of the air increases, it misunderstanding Bernoulli's principle to suggest that this is what it tells us, and the behavior of the curved paper is explained by other reasoning than Bernoulli's principle.|first= Peter|last= Eastwell |title = Bernoulli? Perhaps, but What About Viscosity?|journal = The Science Education Review|volume= 6|issue =1|date = 2007|url =  http://d1vdx9ifs4n5d7.cloudfront.net/s3fs-public/davidson_images/eastwell-bernoulli.pdf}}</ref><ref>"Make a strip of writing paper about 5&nbsp;cm × 25&nbsp;cm. Hold it in front of your lips so that it hangs out and down making a convex upward surface. When you blow across the top of the paper, it rises. Many books attribute this to the lowering of the air pressure on top solely to the Bernoulli effect. Now use your fingers to form the paper into a curve that it is slightly concave upward along its whole length and again blow along the top of this strip. The paper now bends downward...an often-cited experiment, which is usually taken as demonstrating the common explanation of lift, does not do so..." Jef Raskin ''Coanda Effect: Understanding Why Wings Work'' http://karmak.org/archive/2003/02/coanda_effect.html</ref>  A third problem is that it is false to make a connection between the flow on the two sides of the paper using Bernoulli’s equation since the air above and below are ''different'' flow fields and Bernoulli's principle only applies within a flow field.<ref name=Babinsky2>"Blowing over a piece of paper does not demonstrate Bernoulli’s equation. While it is true that a curved paper lifts when flow is applied on one side, this is not because air is moving at different speeds on the two sides... ''It is false to make a connection between the flow on the two sides of the paper using Bernoulli’s equation.''" Holger Babinsky How Do Wings Work ''Physics Education 38(6) http://iopscience.iop.org/0031-9120/38/6/001/pdf/pe3_6_001.pdf</ref><ref>{{cite journal|quote = An explanation based on Bernoulli’s principle is not applicable to this situation, because this principle has nothing to say about the interaction of air masses having different speeds... Also, while Bernoulli’s principle allows us to compare fluid speeds and pressures along a single streamline and... along two different streamlines that originate under identical fluid conditions, using Bernoulli’s principle to compare the air above and below the curved paper in Figure 1 is nonsensical; in this case, there aren’t any streamlines at all below the paper!|first= Peter|last= Eastwell |title =Bernoulli? Perhaps, but What About Viscosity?|journal = The Science Education Review |volume =6|issue =1|date=2007|url = http://d1vdx9ifs4n5d7.cloudfront.net/s3fs-public/davidson_images/eastwell-bernoulli.pdf}}</ref><ref>"The well-known demonstration of the phenomenon of lift by means of lifting a page cantilevered in one’s hand by blowing horizontally along it is probably more a demonstration of the forces inherent in the Coanda effect than a demonstration of Bernoulli’s law; for, here, an air jet issues from the mouth and attaches to a curved (and, in this case pliable) surface. The upper edge is a complicated vortex-laden mixing layer and the distant flow is quiescent, so that Bernoulli’s law is hardly applicable."  David Auerbach ''Why Aircraft Fly'' European Journal of Physics Vol 21 p 295 http://iopscience.iop.org/0143-0807/21/4/302/pdf/0143-0807_21_4_302.pdf </ref><ref>"Millions of children in science classes are being asked to blow over curved pieces of paper and observe that the paper "lifts"... They are then asked to believe that Bernoulli's theorem is responsible... Unfortunately, the "dynamic lift" involved...is not properly explained by Bernoulli's theorem." Norman F. Smith "Bernoulli and Newton in Fluid Mechanics" ''The Physics Teacher Nov 1972''</ref>

As the wording of the principle can change its implications, stating the principle correctly is important.<ref>"Bernoulli’s principle is very easy to understand provided the principle is correctly stated. However, we must be careful, because seemingly-small changes in the wording can lead to completely wrong conclusions." ''See How It Flies'' John S. Denker http://www.av8n.com/how/htm/airfoils.html#sec-bernoulli</ref>  What Bernoulli's principle actually says is that within a flow of constant energy, when fluid flows through a region of lower pressure it speeds up and vice versa.<ref>"A complete statement of Bernoulli's Theorem is as follows: "In a flow where no energy is being added or taken away, the sum of its various energies is a constant: consequently where the velocity increasees the pressure decreases and vice versa."" Norman F. Smith ''Bernoulli, Newton and Dynamic Lift Part I'' School Science and Mathematics Vol 73 Issue 3 http://onlinelibrary.wiley.com/doi/10.1111/j.1949-8594.1973.tb08998.x/pdf</ref> Thus, Bernoulli's principle concerns itself with ''changes'' in speed and ''changes'' in pressure ''within'' a flow field. It cannot be used to compare different flow fields.

A correct explanation of why the paper rises would observe that the [[plume (hydrodynamics)|plume]] follows the curve of the paper and that a curved streamline will develop a pressure gradient perpendicular to the direction of flow, with the lower pressure on the inside of the curve.<ref>"...if a streamline is curved, there must be a pressure gradient across the streamline, with the pressure increasing in the direction away from the centre of curvature." Babinsky  http://iopscience.iop.org/0031-9120/38/6/001/pdf/pe3_6_001.pdf</ref><ref>"The curved paper turns the stream of air downward, and this action produces the lift reaction that lifts the paper." Norman F. Smith ''Bernoulli, Newton, and Dynamic Lift Part II'' School Science and Mathematics vol 73 Issue 4 pg 333 http://onlinelibrary.wiley.com/doi/10.1111/j.1949-8594.1973.tb09040.x/pdf</ref><ref>"The curved surface of the tongue creates unequal air pressure and a lifting action. ... Lift is caused by air moving over a curved surface."  ''AERONAUTICS An Educator’s Guide with Activities in Science, Mathematics, and Technology Education'' by NASA pg 26 http://www.nasa.gov/pdf/58152main_Aeronautics.Educator.pdf</ref><ref>"Viscosity causes the breath to follow the curved surface, Newton's first law says there a force on the air and Newton’s third law says there is an equal and opposite force on the paper. Momentum transfer lifts the strip. The reduction in pressure acting on the top surface of the piece of paper causes the paper to rise."  ''The Newtonian Description of Lift of a Wing'' [[David F. Anderson]] & Scott Eberhardt pg 12 http://www.integener.com/IE110522Anderson&EberhardtPaperOnLift0902.pdf</ref> Bernoulli's principle predicts that the decrease in pressure is associated with an increase in speed, i.e. that as the air passes over the paper it speeds up and moves faster than it was moving when it left the demonstrator's mouth.  But this is not apparent from the demonstration.<ref>'"Demonstrations" of Bernoulli's principle are often given as demonstrations of the physics of lift. They are truly demonstrations of lift, but certainly not of Bernoulli's principle.' David F Anderson & Scott Eberhardt ''Understanding Flight'' pg 229 https://books.google.com/books?id=52Hfn7uEGSoC&pg=PA229</ref><ref>"As an example, take the misleading experiment most often used to "demonstrate" Bernoulli's principle. Hold a piece of paper so that it curves over your finger, then blow across the top. The paper will rise. However most people do not realize that the paper would ''not'' rise if it were flat, even though you are blowing air across the top of it at a furious rate. Bernoulli's principle does not apply directly in this case. This is because the air on the two sides of the paper did not start out from the same source. The air on the bottom is ambient air from the room, but the air on the top came from your mouth where you actually increased its speed without decreasing its pressure by forcing it out of your mouth. As a result the air on both sides of the flat paper actually has the same pressure, even though the air on the top is moving faster. The reason that a curved piece of paper does rise is that the air from your mouth speeds up even more as it follows the curve of the paper, which in turn lowers the pressure according to Bernoulli." From The Aeronautics File By Max Feil https://www.mat.uc.pt/~pedro/ncientificos/artigos/aeronauticsfile1.ps {{webarchive |url=https://web.archive.org/web/20150517081630/https://www.mat.uc.pt/~pedro/ncientificos/artigos/aeronauticsfile1.ps |date=May 17, 2015 }}
</ref><ref>"Some people blow over a sheet of paper to demonstrate that the accelerated air over the sheet results in a lower pressure. They are wrong with their explanation. The sheet of paper goes up because it deflects the air, by the Coanda effect, and that deflection is the cause of the force lifting the sheet. To prove they are wrong I use the following experiment:
If the sheet of paper is pre bend the other way by first rolling it, and if you blow over it than, it goes down. This is because the air is deflected the other way.
Airspeed is still higher above the sheet, so that is not causing the lower pressure." Pim Geurts. sailtheory.com    http://www.sailtheory.com/experiments.html</ref>

Other common classroom demonstrations, such as blowing between two suspended spheres, inflating a large bag, or suspending a ball in an airstream are sometimes explained in a similarly misleading manner by saying "faster moving air has lower pressure".<ref>"Finally, let’s go back to the initial example of a ball levitating in a jet of air. The naive explanation for the stability of the ball in the air stream, 'because pressure in the jet is lower than pressure in the surrounding atmosphere,' is clearly incorrect. The static pressure in the free air jet is the same as the pressure in the surrounding atmosphere..." Martin Kamela ''Thinking About Bernoulli'' The Physics Teacher Vol. 45, September 2007 http://tpt.aapt.org/resource/1/phteah/v45/i6/p379_s1</ref><ref>"Aysmmetrical flow (not Bernoulli's theorem) also explains lift on the ping-pong ball or beach ball that floats so mysteriously in the tilted vacuum cleaner exhaust..." Norman F. Smith, ''Bernoulli and Newton in Fluid Mechanics" The Physics Teacher Nov 1972 p 455</ref><ref>"Bernoulli’s theorem is often obscured by demonstrations involving non-Bernoulli forces. For example, a ball may be supported on an upward jet of air or water, because any fluid (the air and water) has viscosity, which retards the slippage of one part of the fluid moving past another part of the fluid." {{cite web |title=The Bernoulli Conundrum |first=Robert P. |last=Bauman |publisher=Professor of Physics Emeritus, University of Alabama at Birmingham |url=http://www.introphysics.info/Papers/BernoulliConundrumWS.pdf |accessdate=June 25, 2012 |url-status=dead |archiveurl=https://web.archive.org/web/20120225115505/http://www.introphysics.info/Papers/BernoulliConundrumWS.pdf |archivedate=February 25, 2012 }}</ref><ref>"In a demonstration sometimes wrongly described as showing lift due to pressure reduction in moving air or pressure reduction due to flow path restriction, a ball or balloon is suspended by a jet of air." {{cite web |first=Gale M. |last=Craig |title=Physical Principles of Winged Flight |url=http://www.rcgroups.com/forums/showatt.php?attachmentid=5305482  |accessdate=March 31, 2016}}</ref><ref>"A second example is the confinement of a ping-pong ball in the vertical exhaust from a hair dryer. We are told that this is a demonstration of Bernoulli's principle. But, we now know that the exhaust does not have a lower value of ps. Again, it is momentum transfer that keeps the ball in the airflow. When the ball gets near the edge of the exhaust there is an asymmetric flow around the ball, which pushes it away from the edge of the flow. The same is true when one blows between two ping-pong balls hanging on strings." Anderson & Eberhardt ''The Newtonian Description of Lift on a Wing'' http://lss.fnal.gov/archive/2001/pub/Pub-01-036-E.pdf</ref><ref>"This demonstration is often incorrectly explained using the Bernoulli principle.  According to the INCORRECT explanation, the air flow is faster in the region between the sheets, thus creating a lower pressure compared with the quiet air on the outside of the sheets."  {{cite web |title=Thin Metal Sheets – Coanda Effect |publisher=University of Maryland – Physics Lecture-Demonstration Facility |url=http://www.physics.umd.edu/lecdem/services/demos/demosf5/f5-03.htm |accessdate=October 23, 2012 |url-status=dead |archiveurl=https://web.archive.org/web/20120623121737/http://www.physics.umd.edu/lecdem/services/demos/demosf5/f5-03.htm |archivedate=June 23, 2012 }}</ref><ref>"Although the Bernoulli effect is often used to explain this demonstration, and one manufacturer sells the material for this demonstration as "Bernoulli bags," it cannot be explained by the Bernoulli effect, but rather by the process of entrainment." {{cite web |title=Answer #256 |publisher=University of Maryland – Physics Lecture-Demonstration Facility |url=http://www.physics.umd.edu/deptinfo/facilities/lecdem/services/QOTW/arch13/a256.htm |accessdate=December 9, 2014 |url-status=dead |archiveurl=https://web.archive.org/web/20141213125404/http://www.physics.umd.edu/deptinfo/facilities/lecdem/services/QOTW/arch13/a256.htm |archivedate=December 13, 2014 }}</ref>

== See also ==
* [[Daniel Bernoulli]]
* [[Coandă effect]]
* [[Euler equations (fluid dynamics)|Euler equations]] – for the flow of an [[Viscosity|inviscid]] fluid
* [[Hydraulics]] – applied fluid mechanics for liquids
* [[Navier–Stokes equations]] – for the flow of a [[viscosity|viscous]] fluid
* [[Fluid dynamics#Terminology in fluid dynamics|Terminology in fluid dynamics]]
* [[Torricelli's law]] – a special case of Bernoulli's principle

== Notes==
{{Notelist}}
== References ==
{{reflist|2}}

== External links ==
{{commons category|Bernoulli's principle}}
* [http://www.fxsolver.com/solve/share/PbQFI7GERDpBd0fNakKRzA==/ Bernoulli equation calculator]
* [http://mysite.du.edu/~jcalvert/tech/fluids/bernoul.htm Denver University &ndash; Bernoulli's equation and pressure measurement]
* [https://web.archive.org/web/20080201073117/http://www.millersville.edu/~jdooley/macro/macrohyp/eulerap/eulap.htm Millersville University – Applications of Euler's equation]
* [http://www.grc.nasa.gov/WWW/K-12/airplane/bga.html NASA – Beginner's guide to aerodynamics]
* [http://user.uni-frankfurt.de/~weltner/Misinterpretations%20of%20Bernoullis%20Law%202011%20internet.pdf Misinterpretations of Bernoulli's equation – Weltner and Ingelman-Sundberg]

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